C CC l all aa s ss s ss i cii cc a aa l ll M MM e Cee cCC lcc ashlahlah siassassa cnicnicn ai caiai l lclc Ms MsMs e c ec ec hG hGhG a oaa noo nn il cdilil s stcs dscs ds PPEPAEAERARSRSOSONON NN NE NCWECWEl aCWlI aNsIl aNssTIiNssETcisRETcaiRENcalR NalAM NlAMT eAMITecOITecOhINcOhaNAhanNAaLniAc LniEcs LiDEcs DEIsTDIITOIITOINONN Ge Gte Gte oldin oldinoldin GGoGoldoldsldtsetseitnein i n S aS aSfkafokfok o P Po Pooooleolele s te Sste Sste S CClaClaslassissciscaTichaTl haiTlMr hidlMr eidM reEcd dEech dEcihatdiihantoiiantioncinioncsincss inainaina f Safkko P Saffko Saffko GGoGoldoldsldtseTtseihTtneihiTn rih idn r id r SE d aSdE aSdEfiktadfiikotofiikoto ni oo n n P Po Pooooleolele ookPkP oooo o o o Pole Ple Ple oo o leThol Tol T eheh i Tr i i hdTrTr hdhd ird Eir ir dEdE d E d d diEE tdidi iitt tiooitiioitiio noo nnn nn ISBN 978-1-29202-655-8 ISBINS B9N7 89-718--219-220922-0625-56-585-8 9 781292 026558 9 7981728912290226052568558 www.pearson.com/uk wwwww.pwe.apresaorns.ocon.mco/umk/uk 9781292026558_CV_Final.indd 1 7/2/13 11:51 AM 97819279821022962505286_5C58V__CFVin_aFl.iinnadld.i n d1d 1 7/2/173/2 / 1131 : 5 111 A:5M1 AM CLASSICAL MECHANICS Pearson New International Edition Classical Mechanics Goldstein Safko Poole Third Edition PEARSON Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk © Pearson Education Limited 2014 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affi liation with or endorsement of this book by such owners. PEARSON ISBN 10: 1-292-02655-3 ISBN 13: 978-1-292-02655-8 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Printed in the United States of America Contents Survey of the Elementary Principles 1 1.1 Mechanics of a Particle 1 1.2 Mechanics of a System of Particles 5 1.3 Constraints 12 1.4 D' Alembert's Principle and Lagrange's Equations 16 1.5 Velocity-Dependent Potentials and the Dissipation Function 22 1.6 Simple Applications of the Lagrangian Formulation 24 Variational Principles and Lagrange's Equations 34 2.1 Hamilton's Principle 34 2.2 Some Techniques of the Calculus of Variations 36 2.3 Derivation of Lagrange's Equations from Hamilton's Principle 44 2.4 Extending Hamilton's Principle to Systems with Constraints 45 2.5 Advantages of a Variational Principle Formulation 51 2.6 Conservation Theorems and Symmetry Properties 54 2.7 Energy Function and the Conservation of Energy 60 The Central Force Problem 70 3.1 Reduction to the Equivalent One-Body Problem 70 3.2 The Equations of Motion and First Integrals 72 3.3 The Equivalent One-Dimensional Problem, and Classification of Orbits 76 3.4 The Virial Theorem 83 3.5 The Differential Equation for the Orbit, and Integrable Power-Law Potentials 86 3.6 Conditions for Closed Orbits (Bertrand's Theorem) 89 3.7 The Kepler Problem: Inverse-Square Law of Force 92 3.8 The Motion in Time in the Kepler Problem 98 3.9 The Laplace-Runge-Lenz Vector 102 3.10 Scattering in a Central Force Field 106 3.11 Transformation of the Scattering Problem to Laboratory Coordinates 114 3.12 The Three-Body Problem 121 Contents The Kinematics of Rigid Body Motion 134 4.1 The Independent Coordinates of a Rigid Body 134 4.2 Orthogonal Transformations 139 4.3 Formal Properties of the Transformation Matrix 144 4.4 The Euler Angles 150 4.5 The Cayley-Klein Parameters and Related Quantities 154 4.6 Euler's Theorem on the Motion of a Rigid Body 155 4.7 Finite Rotations 161 4.8 Infinitesimal Rotations 163 4.9 Rate of Change of a Vector 171 4.10 The Coriolis Effect 174 The Rigid Body Equations of Motion 184 5.1 Angular Momentum and Kinetic Energy of Motion about a Point 184 5.2 Tensors 188 5.3 The Inertia Tensor and the Moment of Inertia 191 5.4 The Eigenvalues of the Inertia Tensor and the Principal Axis Transformation 195 5.5 Solving Rigid Body Problems and the Euler Equations of Motion 198 5.6 Torque-free Motion of a Rigid Body 200 5.7 The Heavy Symmetrical Top with One Point Fixed 208 5.8 Precession of the Equinoxes and of Satellite Orbits 223 5.9 Precession of Systems of Charges in a Magnetic Field 230 Oscillations 238 6.1 Formulation of the Problem 238 6.2 The Eigenvalue Equation and the Principal Axis Transformation 241 6.3 Frequencies of Free Vibration, and Normal Coordinates 250 6.4 Free Vibrations of a Linear Triatomic Molecule 253 6.5 Forced Vibrations and the Effect of Dissipative Forces 259 6.6 Beyond Small Oscillations: The Damped Driven Pendulum and the Josephson Junction 265 The Classical Mechanics of the Special Theory of Relativity 276 7.1 Basic Postulates of the Special Theory 277 7.2 Lorentz Transformations 280 7.3 Velocity Addition and Thomas Precession 282 7.4 Vectors and the Metric Tensor 286 Contents vii 7.5 1-Forms and Tensors 289 7.6 Forces in the Special Theory; Electromagnetism 297 7.7 Relativistic Kinematics of Collisions and Many-Particle Systems 300 7.8 Relativistic Angular Momentum 309 7.9 The Lagrangian Formulation of Relativistic Mechanics 312 7.10 Covariant Lagrangian Formulations 318 7.11 Introduction to the General Theory of Relativity 324 8 ■ The Hamilton Equations of Motion 334 8.1 Legendre Transformations and the Hamilton Equations of Motion 334 8.2 Cyclic Coordinates and Conservation Theorems 343 8.3 Routh's Procedure 347 8.4 The Hamiltonian Formulation of Relativistic Mechanics 349 8.5 Derivation of Hamilton's Equations from a Variational Principle 353 8.6 The Principle of Least Action 356 9 ■ Canonical Transformations 368 9.1 The Equations of Canonical Transformation 368 9.2 Examples of Canonical Transformations 375 9.3 The Harmonic Oscillator 377 9.4 The Symplectic Approach to Canonical Transformations 381 9.5 Poisson Brackets and Other Canonical Invariants 388 9.6 Equations of Motion, Infinitesimal Canonical Transformations, and Conservation Theorems in the Poisson Bracket Formulation 396 9.7 The Angular Momentum Poisson Bracket Relations 408 9.8 Symmetry Groups of Mechanical Systems 412 9.9 Liouville's Theorem 419 10 ■ Hamilton-Jacobi Theory and Action-Angle Variables 430 10.1 The Hamilton-Jacobi Equation for Hamilton's Principal Function 430 10.2 The Harmonic Oscillator Problem as an Example of the Hamilton-Jacobi Method 434 10.3 The Hamilton-Jacobi Equation for Hamilton's Characteristic Function 440 10.4 Separation of Variables in the Hamilton-Jacobi Equation 444 10.5 Ignorable Coordinates and the Kepler Problem 445 10.6 Action-angle Variables in Systems of One Degree of Freedom 452 viii Contents 10.7 Action-Angle Variables for Completely Separable Systems 457 10.8 The Kepler Problem in Action-angle Variables 466 11 ■ Classical Chaos 483 11.1 Periodic Motion 484 11.2 Perturbations and the Kolmogorov-Arnold-Moser Theorem 487 11.3 Attractors 489 11.4 Chaotic Trajectories and Liapunov Exponents 491 11.5 Poincare Maps 494 11.6 Henon-Heiles Hamiltonian 496 11.7 Bifurcations, Driven-damped Harmonic Oscillator, and Parametric Resonance 505 11.8 The Logistic Equation 509 11.9 Fractals and Dimensionality 516 12 ■ Canonical Perturbation Theory 526 12.1 Introduction 526 12.2 Time-dependent Perturbation Theory 527 12.3 Illustrations of Time-dependent Perturbation Theory 533 12.4 Time-independent Perturbation Theory 541 12.5 Adiabatic Invariants 549 13 ■ Introduction to the Lagrangian and Hamiltonian Formulations for Continuous Systems and Fields 558 13.1 The Transition from a Discrete to a Continuous System 558 13.2 The Lagrangian Formulation for Continuous Systems 561 13.3 The Stress-energy Tensor and Conservation Theorems 566 13.4 Hamiltonian Formulation 572 13.5 Relativistic Field Theory 577 13.6 Examples of Relativistic Field Theories 583 13.7 Noether's Theorem 589 Appendix A ■ Euler Angles in Alternate Conventions and Cayley-Klein Parameters 601 Appendix B ■ Groups and Algebras 605 Selected Bibliography 617 Author Index 623 Subject index 625 Preface to the Third Edition The first edition of this text appeared in 1950, and it was so well received that it went through a second printing the very next year. Throughout the next three decades it maintained its position as the acknowledged standard text for the intro­ ductory Classical Mechanics course in graduate level physics curricula through­ out the United States, and in many other countries around the world. Some major institutions also used it for senior level undergraduate Mechanics. Thirty years later, in 1980, a second edition appeared which was "a through-going revision of the first edition." The preface to the second edition contains the following state­ ment: "I have tried to retain, as much as possible, the advantages of the first edition while taking into account the developments of the subject itself, its position in the curriculum, and its applications to other fields." This is the philosophy which has guided the preparation of this third edition twenty more years later. The second edition introduced one additional chapter on Perturbation Theory, and changed the ordering of the chapter on Small Oscillations. In addition it added a significant amount of new material which increased the number of pages by about 68%. This third edition adds still one more new chapter on Nonlinear Dy­ namics or Chaos, but counterbalances this by reducing the amount of material in several of the other chapters, by shortening the space allocated to appendices, by considerably reducing the bibliography, and by omitting the long lists of symbols. Thus the third edition is comparable in size to the second. In the chapter on relativity we have abandoned the complex Minkowski space in favor of the now standard real metric. Two of the authors prefer the complex metric because of its pedagogical advantages (HG) and because it fits in well with Clifford Algebra formulations of Physics (CPP), but the desire to prepare students who can easily move forward into other areas of theory such as field theory and general relativity dominated over personal preferences. Some modern notation such as 1-forms, mapping and the wedge product is introduced in this chapter. The chapter on Chaos is a necessary addition because of the current interest in nonlinear dynamics which has begun to play a significant role in applications of classical dynamics. The majority of classical mechanics problems and appli­ cations in the real world include nonlinearities, and it is important for the student to have a grasp of the complexities involved, and of the new properties that can emerge. It is also important to realize the role of fractal dimensionality in chaos. New sections have been added and others combined or eliminated here and there throughout the book, with the omissions to a great extent motivated by the desire not to extend the overall length beyond that of the second edition. A section ix